Remark 2.8 
If a length function \(\ell \)  is symmetric and separates points, then \(d(g,h)=\ell (g^{-1}h)\)  determines a left-invariant distance. Thus, under the hypotheses of Theorem REF  we obtain the left-invariant distance associated with a maximal length function exponentially distorted on \(N^{\prime }\) . Hence there is a family of equivalence classes (up to quasi-isometry) of metric spaces parameterized by integral subgroups \(N^{\prime }\)  such that \(E\subset N^{\prime }\subset N\) . By Proposition REF , representatives of two different classes are not pairwise quasi-isometric. The family has two extreme cases: \(N^{\prime }=E\)  (the biggest) and \(N^{\prime }=N\)  (the smallest). The biggest class contains all the distances associated with word length functions on \(G\)  and also with left invariant Riemannian metrics; see, e.g., [1]}). So in the case when \(E\ne N\)  we have left-invariant structures of metric space on \(G\)  that do not correspond to Riemannian metrics.
